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extropalopakettle
No offense, but....

Posted: Wed Aug 24, 2011 5:01 am    Post subject: 81

BraveHat wrote:
 extro wrote: No. Why must an object have width, length, depth, or any size at all? Do quarks have a size? I'm not sure ... they're usually described as occupying a certain minuscule volume, but I think this has something to do with Heisenberg Uncertainty principle, wherein they can't really be said to be at a particular location. The real physical universe is certainly not like what our intuition says is possible. An object can be dimensionless ( and it's location can be distributed over a certain volume).

I'm really not understanding that concept, mainly because I don't know a lot about quarks. But. How can they possibly take up no volume??? If they take up absolutely no volume, what would it even mean for them to exist?

You seem to have a lot of arbitrary axioms about physical reality (like something existing necessitates it having volume).

Why should something existing imply it has volume?

Elementary particles may indeed be points, 0-dimensional. They move about, interact with other particles, are affected by forces, and perhaps create forces. They can be detected, manipulated, studied, their behavior in different situations predicted.

BraveHat wrote:
extro wrote:
extro wrote:
 Antrax wrote: Let's say you pick a random integer with uniform probability.

Not possible. The probabilities have to add up to 1.

Again - not possible. An infinite number of equal non-zero probabilities can't add to 1 (nor can an infinite number of zero probabilities).

Here's where I think we agree, which is that it isn't possible to pick a random integer with uniform probability, but more along the lines of why Zag thinks so, namely, that I'm more likely to pick an integer with less than a billion digits.

Terrible argument. You're not more likely to pick an integer with less than a billion digits if you pick an integer randomly with uniform probability. Is your argument "I, BraveHat, don't know how to do it, therefore it's impossible"?

 BraveHat wrote: I think this is also another candidate for the argument proving the Universe is not infinite in space.

How so?

 BraveHat wrote: (btw, even though I don't personally believe that any Universe could be infinite in space, I never actually claimed that in these discussions, so I'm not sure why you think I'm arguing that. I'm only claiming that my intuition is that the Universe (meaning the one we're in) is not infinite)

I do know we were earlier discussing the impossibility of a universe having an infinite past (no beginning). Not this universe, but perhaps a prior one.

And your first post starting this thread seemed to suggest you were looking for arguments not based on particular scientific observations (which suggest an infinite universe), but "good arguments for the impossibility of infinite quantities in physical reality", which to me suggests they'd apply to any reality.

 BraveHat wrote: If the Universe is infinite in space, then the chances that the Earth could have formed within in any particular block of space is 0. Therefore, before it was formed, the Earth had no chance of forming within in any particular block of space. Yet, the Earth formed within a particular block of space. Therefore, the Universe is not infinite in space.

There are so many fallacies packed in there, it's like a black hole of fallaciousness.

Google anthropic principle. Imagine, for sake of argument, an infinite universe, filled with an infinite number of galaxies, stars, planets. And on maybe only one, intelligent life evolves, and that life form calls their planet "Earth". Saying it's unlikely to happen here is like saying that of the 6 billion human skins on the planet, it's unlikely your body would be inside yours. A 1 in 6 billion chance. But it happened. And likewise for every other of the 6 billion humans too! Impossible!

Second, probabilities are just estimates in the face of unknowns. When I flip a coin in the air, the probability that it will land heads up is either 1 or 0, if I have the information to calculate its trajectory, etc. Without the information or without bothering to calculate, I estimate 0.5. After it lands, I know whether my estimate was too high or too low.

"before it was formed, the Earth had no chance of forming within in any particular block of space" is a hypothetical estimation by some entity in the face of unknowns.

"If the Universe is infinite in space, then the chances that the Earth could have formed within in any particular block of space is 0." ... There seems to be an unstated assumption here. What is it? Uniform probability of it forming in any block of space?
raekuul
Lives under a bridge & tells stories.

 Posted: Wed Aug 24, 2011 5:09 am    Post subject: 82 ...did the four of us just agree on something?
Thok
Oh, foe, the cursed teeth!

Posted: Wed Aug 24, 2011 12:02 pm    Post subject: 83

 BraveHat wrote: Here's where I think we agree, which is that it isn't possible to pick a random integer with uniform probability, but more along the lines of why Zag thinks so, namely, that I'm more likely to pick an integer with less than a billion digits.

There's two arguments going on here. One shows why a specific algorithm (I just pick a number) isn't uniformly random. The other show why no algorithm can be uniformly random. The latter is more important: without that argument it could be that there's some obscure algorithm that none of us haven't thought of that works, even if "Just pick a number" doesn't work.

This goes back to my "you have to be precise and give definitions to actually do mathematics" comment a page or so ago.

@Extro: yes, 10% of the numbers of the real numbers from [0,1] are in [0,.1]. Of course, percentage is now defined in terms of integrals, and the concept of zero percentage is weirder: one can legitimately claim that 0 percent of the numbers from [0,1] are rational, even though those are most of the numbers you normally use. More generally any countable subset will have zero percentage given how we define percentage in that case.

Probability on uncountable sets is weird, but it gives us useful results.
extro...*
Guest

Posted: Wed Aug 24, 2011 7:33 pm    Post subject: 84

extropalopakettle wrote:
Thok* wrote:
extropalopakettle wrote:
 Antrax wrote: Ten percent of the real numbers from 0 to 1 lies within the region 0 to 0.1, ....

Is that true?

It is for a reasonable definition of a uniform probability on [0,1] ...

I didn't think the fragment of what I quoted from Antrax called for any definition of probability.

 Thok wrote: @Extro: yes, 10% of the numbers of the real numbers from [0,1] are in [0,.1]. Of course, percentage is now defined in terms of integrals, ... Probability on uncountable sets is weird, but it gives us useful results.

How do we keep going from percentages to probability? What is our definition of 'percentage' here? I feel kinda dumb ... studied quite a bit of math, but percentage has always been a ratio times 100. But not here.
Thok
Oh, foe, the cursed teeth!

Posted: Wed Aug 24, 2011 10:51 pm    Post subject: 85

 extro...* wrote: What is our definition of 'percentage' here? I feel kinda dumb ... studied quite a bit of math, but percentage has always been a ratio times 100. But not here.

I'm defining it as probability*100%, with the point that the probability is a proxy for the ratio
(amount of numbers in [0,.1])/(amount of numbers in [0,1])
where amount of numbers is being defined in terms of a uniform probability distribution on [0,1].

I'm not sure what (or maybe even if) you dislike this definition.
extra...*
Guest

Posted: Wed Aug 24, 2011 11:56 pm    Post subject: 86

Thok wrote:
 extro...* wrote: What is our definition of 'percentage' here? I feel kinda dumb ... studied quite a bit of math, but percentage has always been a ratio times 100. But not here.

I'm defining it as probability*100%, with the point that the probability is a proxy for the ratio
(amount of numbers in [0,.1])/(amount of numbers in [0,1])
where amount of numbers is being defined in terms of a uniform probability distribution on [0,1].

I'm not sure what (or maybe even if) you dislike this definition.

Then how do you get that rationals are 0% of the reals in the range 0 to 1? (I understand they're countable, and the irrationals uncountable)

What's the probability of choosing an irrational? Just how does on do so? A countable subset of the irrationals have a finite description (like pi and square root of two). How can you choose the ones that don't?
BraveHat
Last of the Daedalians

Posted: Thu Aug 25, 2011 12:39 am    Post subject: 87

 extro wrote: you seem to have a lot of arbitrary axioms about physical reality (like something existing necessitates it having volume).

I don't like to nitpick, but we are philosophizing, after all. You might have guessed from previous posts of mine (not necessarily in these infinity threads) that I'm a theist. So obviously, I don't believe it's necessary for something to take up volume in order to exist. I believe it's necessary for something to take up volume in order to exist as a physical object.

Now you say that axiom is arbitrary, as if I just kind of came up with it out of the blue. To me, it is self-evident. Am I the only one here who finds it plainly obvious that a minimum requirement for being a physical object to is to take up some kind of volume?

 extro wrote: Why should something existing imply it has volume?
Why should a physical object existing imply it has a volume?

 dictionary.com wrote: Physical: of or pertaining to that which is material

If an object has no volume, there is no place for it's material to actually exist. I don't understand how this is not self-evident to you.

 extro wrote: Elementary particles may indeed be points, 0-dimensional. They move about, interact with other particles, are affected by forces, and perhaps create forces. They can be detected, manipulated, studied, their behavior in different situations predicted.

Perhaps these 0-dimensional objects have no material, then. If they have no material, then by definition, they are not physical. If they do have material, then they need volume for that material to exist in.

extro wrote:
 I wrote: Here's where I think we agree, which is that it isn't possible to pick a random integer with uniform probability, but more along the lines of why Zag thinks so, namely, that I'm more likely to pick an integer with less than a billion digits.

Terrible argument. You're not more likely to pick an integer with less than a billion digits if you pick an integer randomly with uniform probability.

You're reversing my argument. I'm not saying I am more likely to pick an integer with less than a billion digits if I pick an integer randomly with uniform probability. I'm saying it's impossible for me to pick an integer randomly with uniform probability because it's more likely I'll pick an integer with less than a billion digits.

 extro wrote: Is your argument "I, BraveHat, don't know how to do it, therefore it's impossible"?

No. I'm saying it's impossible for anyone to do so, because no matter how high an integer someone can name, the range from that integer to it's opposite is as nothing next to the entire infinite integer line.

 extro wrote: I do know we were earlier discussing the impossibility of a universe having an infinite past (no beginning). Not this universe, but perhaps a prior one.

Yes, and I apologize for not directly continuing that discussion. I didn't want to start a new thread with a direct continuation or people might be confused, so instead I asked a new question to start it off. I didn't want to continue it in the other thread, because it was veering off course from the main topic. I will have to go back and revisit what our discussion was before, but I think I tried to argue that the past itself could not be infinite (which would imply that any Universe or anything at all could not be infinite in the past), but I don't think the focus of my argument was on anything but our universe until this thread, when I started to get off track a little (I think).

 extro wrote: And your first post starting this thread seemed to suggest you were looking for arguments not based on particular scientific observations (which suggest an infinite universe), but "good arguments for the impossibility of infinite quantities in physical reality", which to me suggests they'd apply to any reality.

First as you'll note, I hold that it's impossible for any observations to suggest an infinite universe. The "flat space" argument, like any other, only suggests a vast area of space remaining consistent. But as vast as that area is, it could still be finite, at which point it becomes as nothing next to infinity. So no, I don't believe that any scientific argument can even remotely suggest such a gargantuan concept as an infinite universe.

Secondly, yes, I used the wrong phrase. I apologize for that. I've probably been intermingling the terms "the Universe" and "physical reality". I should be more specific with the question I'm trying to get an answer to. I do think "physical reality" is different from "any reality". There is a spiritual dimension to our existence which isn't necessarily physical. It could be, but I don't believe it is. I think the place where consciousness originates is also possibly not physical. But physical reality, in the sense I mean it, has to do with the material and kinds of material surrounding us and effecting us, such as our Universe.

 extro wrote: I think this is also another candidate for the argument proving the Universe is not infinite in space. How so?

Sorry, I wasn't very clear. By "this", I meant the rule that the probability of an element being chosen from an infinite set with uniform probability is 0. I then went on to use the block of space that the Earth was formed in as the element, and an infinite Universe as the infinite set of those blocks of space:

extro wrote:
 I wrote: If the Universe is infinite in space, then the chances that the Earth could have formed within in any particular block of space is 0. Therefore, before it was formed, the Earth had no chance of forming within in any particular block of space. Yet, the Earth formed within a particular block of space. Therefore, the Universe is not infinite in space.

"before it was formed, the Earth had no chance of forming within in any particular block of space" is a hypothetical estimation by some entity in the face of unknowns.

"If the Universe is infinite in space, then the chances that the Earth could have formed within in any particular block of space is 0." ... There seems to be an unstated assumption here. What is it? Uniform probability of it forming in any block of space?

Your last question should have been your first, as the quote it is responding to was the primary statement which the bulk of the argument was based on. The statement you quoted before that one was just a rewording of the last one. The unstated assumption is, again, that the probability of an element being chosen from an infinite set with uniform probability is 0. Blocks of space are the infinite set.

I'm responding to your responses in a different order to accommodate the curious order you used to respond to mine.

 extro wrote: There are so many fallacies packed in there, it's like a black hole of fallaciousness. Google anthropic principle. Imagine, for sake of argument, an infinite universe, filled with an infinite number of galaxies, stars, planets. And on maybe only one, intelligent life evolves, and that life form calls their planet "Earth". Saying it's unlikely to happen here is like saying that of the 6 billion human skins on the planet, it's unlikely your body would be inside yours. A 1 in 6 billion chance. But it happened. And likewise for every other of the 6 billion humans too! Impossible!

I agree. And I never said it was unlikely to happen here. I said there would be a chance of 0 that it happened here or within any particular block of space, and I based that statement soley on the notion that the probability of an element being chosen from an infinite set with uniform probability is 0. The truth of that notion is the candidate for a proof that there is no infinite Universe.

 extro wrote: Second, probabilities are just estimates in the face of unknowns. When I flip a coin in the air, the probability that it will land heads up is either 1 or 0, if I have the information to calculate its trajectory, etc. Without the information or without bothering to calculate, I estimate 0.5. After it lands, I know whether my estimate was too high or too low.

You make a valid point. I hadn't really examined the argument thoroughly, which is why I nominated the notion of probability as merely a candidate. The argument was only an illustration of that candidacy.

I could say that there is alway some unknown about any event, because there is always a chance that some unknown other event will cause a different outcome. No one would ever be able to say with 100% certainty which way the coin will land, no matter how much information they have about it's path, until the outcome is final. But then this would turn into an argument about free will vs. determinism, and that's a different subject.
_________________
"I am declaring it a terrible tragedy for me to die. You may disagree..." --Antrax
Thok
Oh, foe, the cursed teeth!

Posted: Thu Aug 25, 2011 12:53 am    Post subject: 88

 extra...* wrote: What's the probability of choosing an irrational? Just how does on do so? A countable subset of the irrationals have a finite description (like pi and square root of two). How can you choose the ones that don't?

Throw a dart at the unit line. Pick that number. It doesn't matter that I can't right it down (and with probability 1 I can't write it down.)

To see that the rationals must have probability 0, do the following. Order the rationals so you have a well defined sense of first rational, second rational, etc. You can do this since they are countable.

Pick an interval of length episilon*1/2^n that contains the nth rational. Then we've covered the rationals with intervals of combined length at most epsilon*(1/2+1/4+1/8+...) = epsilon. So the rationals must have probability at most epsilon.

And since epsilon is arbitrary, that means the rationals must have probability 0: if they had probability beta, then we'd run into problems since we could take epsilon = beta/2 in the previous paragraph.

Fortunately, the irrationals have probability 1. In fact, the complement of any countable set has probability 1.

Yes, it's incredibly counterintuitive. So what?
BraveHat
Last of the Daedalians

Posted: Thu Aug 25, 2011 12:54 am    Post subject: 89

 Thok wrote: Basically, you have a problem with how probability is defined on infinite sets. You can either accept that probability is defined differently on a set with infinitely many points to get something that works, or claim there are only a finite number of positions in space. The latter seems weird to me.

I think extro brought up a great point, which is that probability is information-dependant. The more information is revealed about the circumstance of an event, the more the probability of the outcome changes. So, technically, there is no objective probability of anything happening. It all depends on how much information the intelligent agent who is calculating has. So if there is no objective probability, the whole argument falls apart.
_________________
"I am declaring it a terrible tragedy for me to die. You may disagree..." --Antrax
BraveHat
Last of the Daedalians

Posted: Thu Aug 25, 2011 1:07 am    Post subject: 90

 Zag wrote: No one tried to claim that the chances of Earth forming in any particular place is evenly distributed across all of space -- there were regions that were more likely than others.
Yes, but the chances of the conditions that made it more likely occurring at any particular point in infinite space, according to that rule of probability, is 0.

Everything else you said, I pretty much agree with.

 raekuul wrote: The main reason you're not right is that you're assuming that it's impossible for the Earth to form and working from there.

No, one of my conclusions was that the Earth had no chance of forming, but that was soley based on the notion of probability that was being passed around as if it were unchangable. The main reason I'm not right, is because probability is not objective. Probability is entirely dependent on what information is involved in the process of calculation. It is a subjective figure.
_________________
"I am declaring it a terrible tragedy for me to die. You may disagree..." --Antrax
Thok
Oh, foe, the cursed teeth!

Posted: Thu Aug 25, 2011 1:13 am    Post subject: 91

 BraveHat wrote: So if there is no objective probability, the whole argument falls apart.

Meh. Unless you believe that the universe is completely deterministic, there's objective probability. Like I said, throw a dart at a board. It may not be a uniform probability but nobody cares. (If I was in more of a mood for an intellectual fight I'd say measure the quantum state of an atom. I'm not sure you believe in quantum mechanics.)

And the mathematics behind probability exists even if it doesn't model the world perfectly: mathematics is just a bunch of results that follow as a consequence from choosing the appropriate set of rules. The results only need to be intuitive/realistic/common sensical if we're trying to use them to model reality.

And not even then. Reality is surprisingly nonrealistic.
raekuul
Lives under a bridge & tells stories.

 Posted: Thu Aug 25, 2011 1:19 am    Post subject: 92 I think it would be more accurate to say that our understanding of reality is unrealistic, but it comes out to be the same result.
Thok
Oh, foe, the cursed teeth!

 Posted: Thu Aug 25, 2011 1:23 am    Post subject: 93 My version sacrifices truth for humor. "And humor is truth, truth humor", that is all ye know on earth, and all ye need to know.
BraveHat
Last of the Daedalians

 Posted: Thu Aug 25, 2011 1:25 am    Post subject: 94 Of our understanding of reality is unrealistic, then who's to say what's real?_________________"I am declaring it a terrible tragedy for me to die. You may disagree..." --Antrax
Thok
Oh, foe, the cursed teeth!

Posted: Thu Aug 25, 2011 1:37 am    Post subject: 95

 BraveHat wrote: Of our understanding of reality is unrealistic, then who's to say what's real?

I trust our ability to measure reality more than I trust our ability to understand it.

Edit: more precisely, I trust our ability to measure reality more than I trust our ability to interpret it.
BraveHat
Last of the Daedalians

Posted: Thu Aug 25, 2011 1:42 am    Post subject: 96

Quote:
 Thok wrote: Of our understanding of reality is unrealistic, then who's to say what's real?

I trust our ability to measure reality more than I trust our ability to understand it.

Edit: more precisely, I trust our ability to measure reality more than I trust our ability to interpret it.

Why?
_________________
"I am declaring it a terrible tragedy for me to die. You may disagree..." --Antrax
Thok
Oh, foe, the cursed teeth!

Posted: Thu Aug 25, 2011 1:57 am    Post subject: 97

 BraveHat wrote: Why?

Why not? Interpretation is hard and requires overcoming built-in biases that come from seeing mostly special cases. Measuring is easy and requires a ruler.
BraveHat
Last of the Daedalians

Posted: Thu Aug 25, 2011 2:15 am    Post subject: 98

 Thok wrote: Why not? Interpretation is hard and requires overcoming built-in biases that come from seeing mostly special cases. Measuring is easy and requires a ruler.

Fair enough. Now do you think we can improve on our ability to interpret observable phenomena, or do you think we will always be equally inadequate at it? Can we improve on our abilities to overcome built-in biases, or do you think it's more likely they will just be replaced by other equally inhibitive biases?
_________________
"I am declaring it a terrible tragedy for me to die. You may disagree..." --Antrax
Thok
Oh, foe, the cursed teeth!

Posted: Thu Aug 25, 2011 8:29 am    Post subject: 99

 BraveHat wrote: Fair enough. Now do you think we can improve on our ability to interpret observable phenomena

Yes. This is what science is. Even if we replace biases with different biases, hopefully those biases are less incorrect.
exyto...*
Guest

Posted: Thu Aug 25, 2011 12:04 pm    Post subject: 100

 Thok wrote: Throw a dart at the unit line. Pick that number. It doesn't matter that I can't right it down (and with probability 1 I can't write it down.)(

According to BraveHat's reasoning, regarding the Earth having zero probability of forming, the dart cannot strike.
BraveHat
Last of the Daedalians

Posted: Thu Aug 25, 2011 10:13 pm    Post subject: 101

Thok wrote:
 I wrote: Of our understanding of reality is unrealistic, then who's to say what's real?

I trust our ability to measure reality more than I trust our ability to understand it.

Edit: more precisely, I trust our ability to measure reality more than I trust our ability to interpret it.

Sorry to be backing up, but I have another question. Are you saying that the closest we can get to knowing what's real is the results of our measurements? Are those results the best grounding we have in what it is real?
_________________
"I am declaring it a terrible tragedy for me to die. You may disagree..." --Antrax
Zag
Tired of his old title

Posted: Thu Aug 25, 2011 11:20 pm    Post subject: 102

 BraveHat wrote: Are you saying that the closest we can get to knowing what's real is the results of our measurements? Are those results the best grounding we have in what it is real?

Well, there is only that which we perceive with our senses (which is always a form of measuring) and that which we imagine. If we aren't careful to base our theories of what's "real" on the former rather than the latter, then we end up with a definition of reality that is based on what we would like to believe, rather than what there is actual evidence for.

If you haven't got evidence then what you have is wishful thinking, and that has no place in science.

Unless, ... oh, wait, you were talking about faith. I'll say this as respectfully as I can: I understand that you disagree, and I respect your right to do so, but I believe faith and wishful thinking to be the same thing.
raekuul
Lives under a bridge & tells stories.

 Posted: Fri Aug 26, 2011 1:35 am    Post subject: 103 When it comes to science, it all comes down to what you can prove. "God Exists" is a hypothesis that I have drawn my own conclusions for, but when it comes to matters of faith, different people doing the same experiment can get wildly different results.
BraveHat
Last of the Daedalians

Posted: Fri Aug 26, 2011 3:42 am    Post subject: 104

Zag wrote:
 I wrote: Are you saying that the closest we can get to knowing what's real is the results of our measurements? Are those results the best grounding we have in what it is real?

Well, there is only that which we perceive with our senses (which is always a form of measuring) and that which we imagine. If we aren't careful to base our theories of what's "real" on the former rather than the latter, then we end up with a definition of reality that is based on what we would like to believe, rather than what there is actual evidence for.

Are those really the only two? What about intuition? Interpersonal "tells"?

 Zag wrote: If you haven't got evidence then what you have is wishful thinking, and that has no place in science.

What happened to imagination? The two are not equivalent. All we do with the infinite number line is imagine it. We can't actually perceive it with our senses. Yet it is not wishful thinking to have an infinite number line. It seems to be the logical consequence of working with abstract mathematics, whether we wish for it or not.

 Zag wrote: Unless, ... oh, wait, you were talking about faith.

say what now?

 Zag wrote: I'll say this as respectfully as I can: I understand that you disagree, and I respect your right to do so, but I believe faith and wishful thinking to be the same thing.

No I was not talking about faith. Not only is that an entirely different debate, I wasn't even debating! I wasn't trying to suggest that Thok was wrong, I was asking him a question to understand what he means. After he answers it, then I'll weigh the answer to see if I agree with it or if there's something more that I think needs to be said. What evidence do you have that my question was leading (and not only leading, but leading to some discussion about "faith" which was not even discussed once in this thread)???
_________________
"I am declaring it a terrible tragedy for me to die. You may disagree..." --Antrax
BraveHat
Last of the Daedalians

 Posted: Fri Aug 26, 2011 4:03 am    Post subject: 105 If time is infinite, does that mean all things that could possibly happen will eventually happen? If space is infinite, does that mean all things that can possibly exist, exist somewhere?_________________"I am declaring it a terrible tragedy for me to die. You may disagree..." --Antrax
BraveHat
Last of the Daedalians

 Posted: Fri Aug 26, 2011 4:06 am    Post subject: 106 Even if a man, from this point on, had infinite time and infinite patience and an infinite life span with an infinite bill of good health, if he started counting to infinity now, he would never reach infinity. So it is not the case that eventually, he will count to infinity._________________"I am declaring it a terrible tragedy for me to die. You may disagree..." --Antrax
extro...*
Guest

Posted: Fri Aug 26, 2011 8:02 am    Post subject: 107

Thok wrote:
 extra...* wrote: What's the probability of choosing an irrational? Just how does on do so? A countable subset of the irrationals have a finite description (like pi and square root of two). How can you choose the ones that don't?

Throw a dart at the unit line. Pick that number. It doesn't matter that I can't right it down (and with probability 1 I can't write it down.)

Maybe I've been out of the math world too long, but I'm having trouble seeing the usefulness of probability in the context of the impossible. Darts are physical objects, or metaphors, and neither can be thrown at a number line, nor if they could could I measure where it landed except by an approximation which would be rational.

Also, this notion of defining percentage as probability assuming a uniform distribution ... Is this a standard definition? I'm just not sure I see the implications.

 Quote: Pick an interval of length episilon*1/2^n that contains the nth rational. Then we've covered the rationals with intervals of combined length at most epsilon*(1/2+1/4+1/8+...) = epsilon. So the rationals must have probability at most epsilon. And since epsilon is arbitrary, that means the rationals must have probability 0: if they had probability beta, then we'd run into problems since we could take epsilon = beta/2 in the previous paragraph. Fortunately, the irrationals have probability 1. In fact, the complement of any countable set has probability 1. Yes, it's incredibly counterintuitive. So what?

That's actually fairly intuitive, putting aside probability of the real world impossibility of "choosing" most reals.

I actually find something like "the rationals in the range 0 to 0.1 are 10% of the rationals in the range 0 to 1" far less intuitive ... Given they're both infinite quantities of the same cardinality.
Thok*
Guest

Posted: Fri Aug 26, 2011 10:11 am    Post subject: 108

 extro...* wrote: I actually find something like "the rationals in the range 0 to 0.1 are 10% of the rationals in the range 0 to 1" far less intuitive ... Given they're both infinite quantities of the same cardinality.

I don't think anybody has claimed that. (I'm pretty sure that in this thread the above phrase has always been stated with reals where the bolded has rationals.)

 Quote: Darts are physical objects, or metaphors, and neither can be thrown at a number line, nor if they could could I measure where it landed except by an approximation which would be rational.

I mean, the board is a number line (or more likely a plane), unless you want to dispute that a number line is a good model for how to describe physical space.

The point about approximation is exactly why we mostly care about the probability of a dart hitting an interval rather than a point. Our approximation won't even be merely rational: it will be to some roughly predetermined fineness depending on our rulers. Measuring things to even 1/1000 of an inch or centimeter is hard without custom tools.

-----------------------------------------------------------------
@Bravehat: I suspect any response I give you will be unhelpful, since I'm not sure what your question is about. I think my answer "Yes, and some clever quip", but I'm not sure what the clever quip should be.
Nsof
Daedalian Member

Posted: Fri Aug 26, 2011 1:44 pm    Post subject: 109

 BraveHat wrote: If time is infinite, does that mean all things that could possibly happen will eventually happen?
if the set of moments in time is countable and the set of all things that can happen is uncountable then no.
I dont get the second question.

 BraveHat wrote: if he started counting to infinity now, he would never reach infinity. So it is not the case that eventually, he will count to infinity.

I am not sure the terms "count to/reach infinity" are used correctly. infinity is not a number we reach/count to. in this case it mean the man will eventually reach any finite number you give him no matter how big it is.

I am not sure what infinite universe means and also isn't the universe diameter 100 billion light years or something?
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extro...*
Guest

Posted: Sat Aug 27, 2011 11:33 am    Post subject: 110

Thok* wrote:
 extro...* wrote: I actually find something like "the rationals in the range 0 to 0.1 are 10% of the rationals in the range 0 to 1" far less intuitive ... Given they're both infinite quantities of the same cardinality.

I don't think anybody has claimed that. (I'm pretty sure that in this thread the above phrase has always been stated with reals where the bolded has rationals.)

Would it be true of the irrationals (reals, minus rationals)?

Is this definition of percentage in terms of probability standard?
BraveHat
Last of the Daedalians

Posted: Thu Sep 01, 2011 3:37 am    Post subject: 111

 listverse.com wrote: In astrophysics and physical cosmology, Olbers’ paradox is the argument that the darkness of the night sky conflicts with the assumption of an infinite and eternal static universe. It is one of the pieces of evidence for a non-static universe such as the current Big Bang model. The argument is also referred to as the “dark night sky paradox” The paradox states that at any angle from the earth the sight line will end at the surface of a star. To understand this we compare it to standing in a forest of white trees. If at any point the vision of the observer ended at the surface of a tree, wouldn’t the observer only see white? This contradicts the darkness of the night sky and leads many to wonder why we do not see only light from stars in the night sky.

_________________
"I am declaring it a terrible tragedy for me to die. You may disagree..." --Antrax
raekuul
Lives under a bridge & tells stories.

 Posted: Thu Sep 01, 2011 4:25 am    Post subject: 112 Planets are poorly-reflective. Gas Clouds are opaque. There, no more Dark Night Sky paradox.
itisally
Master of Disguise

 Posted: Thu Sep 01, 2011 4:35 am    Post subject: 113 BraveHat, I am not sure that the analogy is a good one because light diffuses at a rate that (forgive my lack of memory here) has something to do with dividing by the cube of the distance to the point of it being unperceivable. Basically it could be infinite because the light that would fill that space is from too far away for me to see. This gets back to the idea that I can only use what I can perceive to make conclusions and that those perceptions are colored by what I expect to see. Whenever I note something that is unexpected I must reevaluate my premises. Is it possible that there is unknowable information that will prevent complete knowlege? I was once told that science is not fact it is only our best guess and a good scientist tries to find ways to prove themselves wrong, while a good philosopher tries to find ways to prove themselves right._________________I could agree with you, but then we would both be wrong.
BraveHat
Last of the Daedalians

Posted: Fri Sep 02, 2011 12:21 am    Post subject: 114

 Thok wrote: @Bravehat: I suspect any response I give you will be unhelpful, since I'm not sure what your question is about. I think my answer "Yes, and some clever quip", but I'm not sure what the clever quip should be.

Oh, just a yes or no is fine. I'm really just looking at the conversation as a discovery process, not really leading anywhere.

 itisally wrote: Basically it could be infinite because the light that would fill that space is from too far away for me to see.

That's pretty interesting. So, supposing that somewhere, beyond the point where we can see light, there was an infinite wall of stars all bordering each other to form an infinite plane. You're saying we would still see only black from our perspective? (that, is of course, until the light reached our eyes, and then we'd be screwed)

 itisally wrote: I was once told that science is not fact it is only our best guess and a good scientist tries to find ways to prove themselves wrong, while a good philosopher tries to find ways to prove themselves right.

Oooo,never heard that one. I like it.
_________________
"I am declaring it a terrible tragedy for me to die. You may disagree..." --Antrax
extropalopakettle
No offense, but....

Posted: Sat Sep 03, 2011 7:40 am    Post subject: 115

BraveHat wrote:
 itisally wrote: Basically it could be infinite because the light that would fill that space is from too far away for me to see.

That's pretty interesting. ...

Not just pretty interesting, but also exactly what the quote you posted was suggesting:

BraveHat wrote:
 listverse.com wrote: In astrophysics and physical cosmology, Olbers’ paradox is the argument that the darkness of the night sky conflicts with the assumption of an infinite and eternal static universe. ...

Yes, if the universe always existed, and was always infinite and populated throughout with stars, we'd be flooded with light from recent nearby stars and long dead far away stars. But our universe is young, and the far away stars are not yet visible to us.
raekuul
Lives under a bridge & tells stories.

 Posted: Sat Sep 03, 2011 3:23 pm    Post subject: 116 Actually, the problem I have with Olber's Paradox is that I'm thinking along the lines of the Zeno Paradox. Assume an infinite universe that contains objects that are separated by a finite distance. Which objects are the furthest out? How long will it take for the light from the two most distant objects to reach each other? Where are the objects in relation to our position in the universe?
bgg1996*
Guest

 Posted: Sat Sep 03, 2011 3:37 pm    Post subject: 117 Why can't there just also be an infinite number of gas clouds, planets, and other non-empty space things. Or maybe the empty space itself isn't completely transparent. Or maybe empty space isn't really empty. An infinite amount of light in an infinite amount of space/=/a white sky. In such a universe that "every" sight-line ended in a star, we could still look down and not see a star.
extropalopakettle
No offense, but....

Posted: Sat Sep 03, 2011 3:45 pm    Post subject: 118

 bgg1996* wrote: Why can't there just also be an infinite number of gas clouds, planets, and other non-empty space things.

Wouldn't they get hot enough to emit light if they were absorbing that much light?
bgg1996
BeeGees are awesome!

 Posted: Sat Sep 03, 2011 3:49 pm    Post subject: 119 No, because they are infinite, and can then absorb an infinite amount of light._________________The one member below 18
bgg1996
BeeGees are awesome!

 Posted: Sat Sep 03, 2011 3:49 pm    Post subject: 120 Not to mention black holes._________________The one member below 18
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